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IB Mathematics AHL 5.17 coupled differential equations AI HL Paper 1- Exam Style Questions- New Syllabus

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Question

A dynamic system is represented by the following coupled first-order differential equations:
\(\frac{dx}{dt} = 3x – 2y\)
\(\frac{dy}{dt} = 4x – y\)
The phase portrait below illustrates a trajectory that passes through the coordinates \((1, 0)\) at time \(t=5\).

(a) Based on the characteristics of the phase portrait, identify which of the following could be an eigenvalue for this system.

  • A. \(1\)
  • B. \(2i\)
  • C. \(1+2i\)
  • D. \(-1+2i\)
(b) Use Euler’s method with a constant step size of \(h=0.1\) to estimate the values of \(x\) and \(y\) when \(t=5.5\).

Most-appropriate topic codes:

• AHL 5.17: Phase portrait for the solutions of coupled differential equations — part (a)
• AHL 5.16: Numerical solution of the coupled system by Euler’s method — part (b)
▶️ Answer/Explanation
Detailed solution

(a)
The phase portrait displays a trajectory spiraling away from the origin (a spiral source). This behavior indicates that the system’s eigenvalues must be complex with a positive real part. Therefore, the correct option is C (\(1+2i\)).

(b)
We apply Euler’s method for coupled systems using the initial conditions \(t_0=5, x_0=1, y_0=0\) and step size \(h=0.1\). The iterative formulas are:
\(x_{n+1} = x_n + 0.1(3x_n – 2y_n)\)
\(y_{n+1} = y_n + 0.1(4x_n – y_n)\)

\(n\)\(t\)\(x\)\(y\)
\(0\)\(5\)\(1\)\(0\)
\(1\)\(5.1\)\(1.3\)\(0.4\)
\(2\)\(5.2\)\(1.61\)\(0.88\)
\(3\)\(5.3\)\(1.917\)\(1.436\)
\(4\)\(5.4\)\(2.2049\)\(2.0592\)

For the final step to \(t=5.5\):
\(x = 2.2049 + 0.1(3(2.2049) – 2(2.0592)) \approx 2.45453\dots\)
\(y = 2.0592 + 0.1(4(2.2049) – 2.0592) \approx 2.73524\dots\)

Estimated values at \(t=5.5\):
\(x \approx 2.45\)
\(y \approx 2.74\)

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